State space control methods have been applied successfully to the control of electromagnetic actuators, particularly to the comparatively linear problem of motion control in read-write heads for computer disk drives. Of particular concern, however, are solenoids with two latching positions and a strong spring driving the armature from one latching position to the other. Such “dual-latching” solenoids, of particular use in electromagnetic engine valves, have proved difficult to control. In them, the dominant force comes from a spring system that restores the armature towards a point roughly midway between the two latching positions. Even when driven to saturation, a magnetic yoke cannot pull in and latch the armature starting from a centered rest position. Solenoids of this type must generally be initialized by resonating the armature from side to side several times until it comes close enough to a yoke for capture and latching. Once latched, the armature is released from one side and speeds to the other side, driven mostly by spring force. Magnetic control is effectively lost on release when the armature has passed roughly 20% of its transit distance, while pull-in control becomes effective only in the final 20% or so of travel. To bring such a solenoid up to near saturation and maximum pull across a large armature gap, for example, across 20% of maximum travel, typically requires on the order of one joule of magnetic energy. If the solenoid drive circuit is limited to a moderately high peak power level, for example, two kilowatts for a strong field across a 20% gap, the implication is that about a half millisecond should be required either to build up or break down the magnetic field, as needed to effect a large fractional change in magnetic force. This time figure, about a half millisecond, turns out to be roughly the minimum time to bring magnetic force from zero up to a maximum near saturation, or from that maximum back down to zero. Given a total stop-to-stop solenoid travel time of around three milliseconds, this half-millisecond one-way slew-time figure indicates a severe slew rate constraint for a controlled change of magnetic force. The extent of course correction is therefore severely constrained.
While peak velocities exceeding 3 meters/second are commonly required for sufficient solenoid speed, armature landing impact velocities are desirably held to about 0.03 meters/second or less—only 1% of the peak velocity. Since kinetic energy varies as the square of velocity, a 1% velocity error at landing represents a 0.01% error in kinetic energy, relative to the maximum. By implication, if a low-impact landing strategy were to rely solely on a kinetic energy determination at mid-course, then the energy correction would need to be precise to within 0.01%. Since the trajectory of the armature near landing is unstable and divergent, however, the allowable error in mid-course energy correction needs to be well below 0.01% for open-loop low-impact landing. If such precision is impractical on a laboratory bench, it is impossible in a vibrating engine with turbulent gases swirling past the actuated engine valve. Closed loop control is clearly necessary to control landing energy in a range below 0.01% of peak kinetic energy.
To perform well and land softly under variable operating conditions in an internal combustion engine, dual-latching electromagnetic valve actuators require an “intelligent” closed-loop control process to guide the system trajectory along a narrow landing path, allowing only a few percent of energy deviation before the system strays too far off course for possible correction. Too fast an approach leads to unavoidable impact and bounce. Too slow an approach commonly causes the armature to lose momentum and possibly even reverse direction momentarily, after which an increasing magnetic field overpowers the reversal of motion and pulls the solenoid in for a high-impact landing. An even slower approach results in complete failure to land—even the maximum possible field in deep magnetic saturation cannot reach across the air gap with sufficient strength to bring the armature in against the opposing spring force. These situations are analogous to trajectory control for spacecraft re-entry into Earth's atmosphere from lunar orbit—too steep an entry burns the craft, slightly too shallow an entry causes the craft to bounce off the atmosphere and then burn on too steep a second entry, and an even shallower entry bounces the craft far off into space. Solenoid course corrections must be initiated early, by analogy to exit from lunar orbit. Release from a latching side of a solenoid may require control in order for the opposite, capturing side of the solenoid to bring about successful landing. Consider, for example, where exhaust gas pressures retard the opening of a solenoid-driven valve. The releasing solenoid should reduce its field rapidly after release, to minimize the magnetic retarding force on the departing armature. On the other hand, when pressure in a supercharged intake manifold boosts the energy of an opening intake valve, the releasing solenoid should increase its magnetic strength quickly after release, to reach out and retard the departing armature, removing some of the excess energy.
Under the circumstances just described, generic feedback control schemes are ineffective. The most effective control system embodies specific knowledge of the nonlinear characteristics of the solenoid to be controlled. Effective control requires a built-in description of the range of trajectories that can, under feedback control, be directed to low-impact landing, starting from variable initial energy conditions. By the spaceship analogy, the system must contain a description of the envelope of possible paths that can reach successful landing. The system must be capable of maintaining the system trajectory within the confines of that envelope.
The best available examples from existing control technology in this area fail to meet the challenges just described. Working valve solenoid actuators have been demonstrated, but landing impacts under variable engine operating conditions create noise problems and limit the longevity of solenoid components. The tightest control systems require separate motion sensors for servo feedback, while only one reported “sensorless” control system offers the possibility of multiple trajectory corrections on approach to landing.
In U.S. Pat. No. 6,285,151, Wright and Czimmek describe a sensorless “Method of Compensation for Flux Control of an Electromagnetic Actuator.” Similar material is described in the 2000 SAE Congress paper 2000-01-1225, “Sensorless Control of Electromagnetic Actuators for Variable Valve Train” by Melbert and Koch. In U.S. Pat. No. 6,657,847, Wright and Czimmek further describe an alternative sensorless “Method of Using Inductance for Determining the Position of an Armature in an Electromagnetic Solenoid,” and in U.S. Pat. No. 6,681,728, Peterson, Stefanopoulou, Megli and Haghgooie disclose a similar “Method for Controlling an Electromechanical Actuator for a Fuel Charge Valve.”
Both Wright '847 and Haghgooie '728 view the control problem from the standpoint of being in the right place at the right time and each teaches a method of continuously monitoring velocity, position and current together and adjusting drive voltage each time an error is observed. These methods provide empirical formulas for specific points of course correction in the trajectory of a solenoid moving from one latching position to an opposite latching position, with a goal of low-impact landing with simultaneous magnetic latching. Such systems as these provide some measure of control, but less than is needed for a versatile, quiet-running and long-lasting system. In the words of Wright '847: “Generally, PID (proportional, integral, derivative) control systems can only perfectly compensate a linear system with state variables that are not interactive. Electromagnetic actuators are, however, highly non-linear (and) the state variables are highly interactive.”
In light of the current invention, these methods warrant detailed discussion.
Concerning time and control, Wright '847 explicitly departs from the PID control methods, stating (column 2, lines 41–65) that “. . . there is a need for a true multivariate control system capable of controlling all state variables simultaneously and compensating a nonlinear feedback control system.” Wright '847 goes on to describe a state space whose dimensions are position, velocity, electrical current and time. As shall be shown, a better selection would be to eliminate the dimension of time altogether and to substitute flux linkage for current, resulting in simpler, more linear relationships and improved system stability.
The special significance of three state space dimensions (as opposed to any other number) is the number of time-integration delays between a control input change and response in position. Starting from a control voltage, current and flux linkage vary as the time integral of voltage, so that's one “delay.” Magnetic force and acceleration changes occur with virtually no delay in relation to flux, so the next significant integration delay is going from force and acceleration to velocity. The third integration delay is in going from velocity to position. All solenoids are subject to at least third-order delay. Current controllers can only alter current at rates permitted by their voltage output range. The slew rate for current varies with the maximum volts-per-turn in the winding. Raising the supply voltage calls for higher-voltage transistors and higher instantaneous power capability. Lowering the number of windings causes the solenoid to draw more current, again raising the instantaneous power demand and also increasing power losses from fixed resistances in transistors and circuit board traces. There is a strong economic incentive to design a solenoid system for operation within relatively low slew rate limits. Voltage limiting in current control systems creates a very difficult slew-rate nonlinearity. Applicant's system will be seen to use pre-planned control trajectories within system voltage limits, avoiding slew by design.
Current controllers can only alter current at rates permitted by their voltage output range. The slew rate for current varies with the maximum volts-per-turn in the winding. Raising the supply voltage calls for higher-voltage transistors and higher instantaneous power capability. Lowering the number of windings causes the solenoid to draw more current, again raising the instantaneous power demand and also increasing power losses from fixed resistances in transistors and circuit board traces. There is a strong economic incentive to design a solenoid system for operation within relatively low slew rate limits. Voltage limiting in current control systems creates a very difficult slew-rate nonlinearity. Applicant's system will be seen to use pre-planned control trajectories within system voltage limits, avoiding slew by design.
Considering practical solenoids of the type used to actuate internal combustion engine valves, the spring forces are so high that they easily dominate over the controllable magnetic force across much of the armature's range of travel. As noted in Haghgooie '728, the solenoid controller exerts very low “control authority,” caused both by the dominance of the spring force and by the ‘open’ position of the armature over most of its travel. The solenoid acts primarily as an oscillating spring-mass system whose motion is only under significant controller influence when the armature is very close to one or the other of the two attracting pole faces. Wright '847 states (col. 11, lines 18–22) that “As a rule of thumb, the armature should be close enough to the stator core that the amount of magnetic flux closed through the core is at least equal to the amount of flux that escapes the core.”
In fact, it can be shown that in the region between 20% and 80% of full travel, motion is virtually unperturbed by control action and is governed primarily by the simple harmonic motion of the spring/mass system. As a further important consequence, it is virtually impossible for any controller to significantly influence the overall transit time from armature release to armature recapture. By the time an armature arrives at the final 20% region of significant landing control force, it is too far behind or ahead of schedule for correction, unless its initial kinetic energy was virtually unperturbed by variable operating conditions.
Examining wright '847 in more detail, we find in 130 of FIG. 12 a graph labeled “Soft Landing Position vs. Time.” As is clear throughout wright '847, the armature trajectory is intended to be controlled to track along a single preferred position-versus-time trajectory, such as trajectory 130. Observation, though, shows that in a solenoid system with low control authority, this approach falls short. An armature whose release opens an automotive exhaust valve (for example) will lose energy quickly after release due to a combination of weak magnetic attraction and strong opposing gas forces from out-rushing exhaust. By mid-course, the armature and valve will have a perturbed kinetic energy due to varying launch conditions. No magnetic controller can be expected to cause such variably perturbed trajectories to converge onto a single position-versus-time graph ending at a specific elapsed time after release. Wright '847 calls for the system to do just that. The consequence is that Wright's system can only successfully control armature motions within a very restricted range of energies established shortly after armature release.
As with the teachings of Wright '847, Haghgooie '728 implies a specific, restrictive time schedule for position, as at column 4, lines 20–25: “In each stage in the operation of the closed-loop controller, the voltage command signal generated by the controller is equal to:Voltage=Ki(idesired−imeasured)+Kx(Xdesired−Xmeasured)+Kv(VdesiredVmeasured)”
The values of three state variables, current, position and velocity, are each subtracted from “desired” control values of current, position and velocity at specific moments in time. Haghgooie '728 describes procedures for a flux initialization stage and for a landing stage, but it makes clear that in both stages, the form of the control equation is the same. Little is said about how said desired values are established, except that closed loop control is employed, that there is a switch from a flux initialization algorithm to a soft landing algorithm, and that (column 4, lines 32–34) “In the preceding equation, Ki, Kx, and Kv are constants that are determined using a known linear quadratic regulator optimization technique (LQR) .” There is no discussion of the three “desired” state variables i, x and v, but clearly they are at least functions of time.
Both Wright '847 and Haghgooie '728 rely on a single “desired” path through state space, wherein every position coordinate along that path is a predetermined function of the time dimension. Note that in Wright '847's FIG. 12, graph 130 defines a target position versus time for a single trajectory path through state space. Graph 132 in that same figure, plotting velocity as a function of position, is derived from position information that is already fully defined by the positions and slopes of graph 130. These two graphs simply represent different views of a single trajectory through state space, rather than information about two or more trajectories. Note further that the proportional and rate signals represented at 136 and 140 of the figure are used to derive the third dimension of the state space, in this case expressed as a target electric current. Both Wright '847 and Haghgooie '728 describe methods for causing a servo-controlled trajectory to attempt to track a single, specific target path, itself described as a function of time.
Because inductance places a practical upper limit on the slew rate of the flux linkage curve, and because a solenoid yoke can only attract and cannot repel an approaching armature, there are very restricted options for achieving the simultaneous arrival of flux at its latching value and velocity at a near-zero landing value exactly when the landing position is reached. As shall be made clear, eliminating the time constraint affords a simpler, more effective method for converging to a target strip of trajectories with a finite width in three-dimensional state space. That strip, defined by a collection of known successful trajectories pre-derived from testing or simulation, permits choosing a trajectory most closely aligned with the solenoid's present state, whenever measured, and gradually steering it to a successful landing, thus avoiding flux slew-rate limiting.
A paper by Peterson, Stefanopoulou, and Wang, “Control of Electromechanical Actuators: Valves Tapping in Rhythm” (in Multidisciplinary Research in Control: The Mohammed Dahleh Symposium 2002. Eds. L. Giarre' and B. Bamieh, Lecture Notes in Control and Information Sciences N. 289, Springer-Verlag, Berlin, 2003, ISBN 3-540-00917-5) describes valve actuator control systems at three levels: a linear controller using different approximate equations for the armature far from and near to the attracting yoke; a nonlinear controller; and the same nonlinear controller enhanced by an inter-cycle adjustment, a learning algorithm achieving a desired low level of impact after several training cycles. Earlier related papers by Peterson, Stefanopoulou and others include the titles “Nonlinear Self-Tuning Control for Soft Landing of an Electromechanical Valve Actuator” (K. Peterson, A. Stefanopoulou, Proceedings of IFAC Mechatronics Conference, November 2002) and “Iterative Learning Control of Electromechanical Camless Valve Actuator.” (K. S. Peterson, A. G. Stefanopoulou, Y. Wang, T. Megli, Proceedings IMECE DSCD 2003-41270.) These papers describe various aspects of a controller that employs frequent sampling of position to infer all the state variables of the system. As the first-mentioned paper (above) states: “The system suffers from low control authority in controlling the armature position from the voltage input throughout the executed motion. The underlying reasons are dynamic during small gaps and static during large gaps.” The low static authority during large gaps, as described above, arises because for most of the travel between 20% and 80% of the travel interval, the force of the mechanical spring substantially exceeds the maximum achievable electromagnetic force. As is emphasized in the invention to be described below, most control of the armature trajectory must be exerted early (below 20% of travel) and late (above 80% of travel) in the transition from open to closed gap, although it becomes necessary to anticipate control forces late in the transition by building up a magnetic flux ahead of time, when forces are weak.
The low dynamic authority during small gaps is explained by Peterson et. al. (in the first-mentioned paper above) in terms of “decreased inductance combined with high back-emf that drives the current to zero exceedingly fast.” This statement may be considered confusing (inductance is not “decreased” but becomes very large at small gaps). At issue is a confusing selection of state variables: position, velocity, and current. A less confusing and mathematically “better behaved” representation to be employed below chooses these state variables: position, velocity, and magnetic flux linkage, that is, the flux that effectively threads the coil windings. For an “ideal” solenoid consisting of an infinitely permeable core material and achieving perfect magnetic closure at the latching point, the mathematical representation of the papers by Peterson et. al. becomes singular at magnetic closure, with current going to zero, inductance and back-emf going to infinity as closure is approached at a finite velocity, and magnetic force becoming infinitely sensitive to the vanishing current.
Returning to the first paper, “High cost and implementation issues preclude the use of sensors to measure all three states. For each of the controllers presented later in Sect. 7 only a position sensor is used, and an observer is implemented to estimate velocity and current. Unfortunately, the observability matrix [CT (CA)T (CA2)T]T where A is from the far model and C=[0 1 0], is ill-conditioned. Therefore one or more states are weakly observable from the position measurement.” This “observer” is a computation to infer the unmeasured state variables, velocity and current, based on incoming position data. To say that the observability matrix is ill-conditioned is to indicate that the computational procedure is error-prone. From the same paper, “The current estimate matches the actual state closely for the initial part of the transition, with the estimation error increasing toward the end. Recall that (20) does not include the saturation region in (6)–(8), thus at the end of the transition when saturation occurs the nonlinear model is not accurate.” This inaccuracy cannot be ignored, because the end transition is the region where the landing trajectory must be fine-tuned for low-impact landing. Furthermore, the statement, “ . . . at the end of the transition when saturation occurs . . . ” is either questionable or implies poor control. Static tests of solenoid force and current at various armature positions generally indicate saturation by a slowing of the increase of force with progressive reduction of the armature-yoke gap. These static tests do not reflect dynamic gap closure, where the magnetic flux linkage changes fractionally by only a small amount while coil current may be reduced by ten-to-one or more as the solenoid closes. As an efficient solenoid closes, the relatively constant magnetic flux linking the winding progressively redistributes to bridge the magnetic gap and become linked to the armature. When the solenoid finally closes, virtually all the winding flux linkage also links the armature portion of the magnetic circuit, producing magnetic force. It is not productive to generate more winding flux than can go through the armature at saturation, for the excess flux linkage will cause a high current and high power consumption with a negligible increase in magnetic force. The solenoid should be designed so that the flux capacity of the armature is similar to or slightly less than the flux capacity of the yoke (recognizing that minimizing armature moving mass drives the designer to keep the armature flux capacity no higher than needed.) The controller, in turn, should both compute and control flux, and should include and utilize a model of saturation flux linkage as a variable function of armature position. Recognizing that saturation boundary, the controller should utilize the full flux allowed within that boundary, as needed, without crossing the boundary and dissipating power unproductively.
Going more deeply into the issue of control in the magnetic end transition region, that region frequently corresponds to an engine valve approaching mechanical closure with gases moving rapidly through the closing valve. As the valve approaches closure and interrupts the gas flow, the pressure differential across the valve will increase abruptly, exerting a force that alters the trajectory and causes either closure with impact, or a bounce away from completion of closure, followed either by delayed closure with impact or by complete failure to close and latch. A control system is needed that can do more than get through an end transition to low-impact landing on a laboratory bench, after many learning cycles. The control system needs to maintain a complete model of the system state variables and be capable of making appropriate course corrections on-the-fly, with conditions varying from one cycle to the next. The system should infer the presence of unexpected gas forces based on deviations of the system trajectory from an expected course through the state space. The system should respond to these deviations not only by way of feedback correction, but also with feed-forward correction, based on a built-in gas dynamic model that anticipates coming changes in gas-dynamic forces using present course deviations. Because of the well-recognized third-order response “sluggishness” of solenoid actuators, a system is needed that accomplishes corrective feed-forward actuation in a short time frame, with only a few computations.
The dynamic models taught in the references by Peterson et. al. include a pair of linear models, one for large magnetic gaps and one for small magnetic gaps, and a nonlinear model, applied during the particularly nonlinear small gap region. The nonlinear model, which slightly outperforms the linear model, employs the following control equation:Vc=K1·v/(γ+z)+K2/(β+z)  1
In Eq. 1, Vc is applied control voltage, z is armature position, v is armature velocity, and the four coefficients K1, K2, γ, and β can be varied to optimize landing. The needed optimization is based on an extremum-seeking mathematical procedure that adjusts the four coefficients and optimizes landings iteratively over several actuation cycles. This process is empirical and does not benefit from a more detailed and mechanistic model of the nonlinear state-space dynamics of the controlled system. On the other hand, detailed models have previously been associated with lengthy computations, incompatible with the design of real-time controllers that must respond to incoming sense signals with control output signals in a short time frame, for example, between tens and hundreds of microseconds in a system to control a trajectory lasting under four milliseconds. What is needed and not provided by earlier articles and patent teachings is a control approach that incorporates a complex dynamic model into a procedure that dictates control responses with few computations.
A feedback trajectory control system taught by Bergstrom in U.S. Pat. No. 6,249,418 uses coil current measurement, as in the system of Wright and Czimmek, but in this case the system infers position and magnetic flux linkage at many sampling points along the trajectory, approaching continuous feedback control. Velocity is computed from changes in position. This system is therefore an example where the chosen state variables are position, velocity, and flux linkage. The system derives all the information for a complete state space description at frequent sampling intervals, without some of the indeterminacies that plague the system reported by Peterson, Stefanopoulou and others in the above three references. In particular, the system by Bergstrom is immune to unexpected magnetic saturation, since the primary control is over magnetic flux.
In one of the approaches taught by Bergstrom, solenoid electric current is measured at frequent intervals while inductive voltage is inferred from supply voltage, from the microprocessor Pulse Width Modulation (PWM) setting or duty cycle, and from knowledge of the solenoid current and circuit resistance. The inductive voltage is integrated over time, starting from a time of known zero or near-zero solenoid flux, to give the coil flux linkage. Solenoid position is determined as a function of the ratio of current to flux linkage, referred to in Bergstrom's patent simply as flux. Once position is determined, comparison of the most recently determined position with one or more previous position values indicates solenoid velocity. From solenoid position and flux linkage, it is possible to compute magnetic force. Combining a knowledge of force with corrective feedback, one can obtain linearized control of force, at least within limits of saturation and slew rate of the magnetic field producing the force. With knowledge of position, velocity, and force, and with actuation control over the rate-of-change of force via PWM control, one might expect to be able to apply well known control theory to the solenoid trajectory, as suggested by Bergstrom. Several problems arise in practice, however, largely related to the “low control authority” described by Peterson, Stefanopoulou and others. In the framework of position, velocity, and flux linkage as state variables, the “control authority” problems consist of low force at large gaps for any attainable flux linkage, and a voltage-limited slew rate of flux linkage, preventing rapid changes in that state variable. A related system boundary is the saturation limit of flux linkage. As will be described in detail below, these boundaries constrain the dynamic trajectory of the system to a narrow path in state space. If the system strays outside this path, it cannot be brought back in time, resulting in high-impact landing and/or failure to latch—the analogy to spacecraft reentry into Earth's atmosphere starting from lunar orbit is recalled. Bergstrom fails to disclose specific methods for maintaining a fast solenoid system trajectory within narrow path boundaries, although the flux control method taught by Bergstrom is a useful start in creating such methods.
Like the system described by Peterson, Stefanopoulou and others, the system by Bergstrom combines information from a single measured variable with knowledge of a second variable, for example, the drive voltage or control voltage, Vc, appearing in Eq. 1. The system by Bergstrom has the advantage that the measured variable, current, is easily measured at the controller circuit board, whereas a direct position measurement requires a sensor at the actuator, in the harsh engine environment. An appropriately robust sensor for this use, as shown in the first-mentioned paper (above) by Peterson et. al., is an eddy current sensor, measuring a position-dependent AC impedance and requiring oscillator and demodulation or detection circuitry. While the two state-variable choices, of (position, velocity, flux-linkage) versus (position, velocity, current), are formally equivalent in that the one description can be transformed into the other, the flux-linkage choice leads readily to simpler expressions, useful in mapping out a comprehensive description of pre-computed state space trajectories, which is a novel basis for the invention to be described below.